India is not dependent on Russia, they can pick and choose from whatever arms supplier they desire (including now also the US), as they have done profusely. Just look at the strange mix of Russian, German and French submarines that forms India's submarine force. It is rather Russia that is seeking to keep India as close as possible as a major ally, especially now that the US has entered the fray of course.
I have little doubt that Russia, like India, views Pakistan as an adversary (they have been on opposite sides in Afghanistan for 30 years or more). So I persist in thinking that Russia had to convince the Indians - actually you mention Monmahan Singh's voyage to Moscow - that the FC-1 would have no effect on the balanced (or rather disbalance) between Pakistan and India.